The player level is advanced by research and building the correct buildings to allow the Operations Center to be upgraded. Leveling up also gives access to different ships and research abilities. Here are the notable milestones for various ships.
Level 1:
Realta (1* Common Explorer)
Level 5:
Orion Corvette (1* Common Battleship)
Level 7:
Fortunate (1* Comon Mining Ship)
Level 9:
Astronautics (Research Efficiency)
Level 10:
Refinery
Player vs Player (Ships only)
Level 12:
Turas (2* Comon Explorer)
Envoy (2* Common Mining Ship)
Jellyfish (2* Rare Explorer)
Level 14:
Talla (2* Common Battleship)
Level 15:
Player vs Player (Base Hits)
Level 17:
Franklin (3* Common Explorer - Swarm)
Level 18:
North Star (3* Rare Mining Ship)
Ship Hanger (Ship count, Hull Health)
Level 20:
Kehra (2* Common Interceptor)
Scrapyard
Level 22:
Vahklas (3* Common Explorer)
Level 23:
D'Vor (Latinum Mining Ship)
Armada Control Center (Launch Armadas)
Engine Technology Lab (Dodge, Interceptor Shield)
Foundry (Armor, Battleship Shield)
Science Lab (Shield Deflection, Explorer Shield)
Level 24:
Kumari (3* Common Battleship)
Below is a chart from LCARS showing the various building requirements by level:
